no they would go on the webserver, along with all your html files. except for the free counter types (like statcounter), in which you just include a snippet of their code in your html files.
im not really sure what you mean. the code would reside on the webserver (in the actual webpages) with the visit count for everyone to see, unless you tell statcounter to count, but not display a counter.
I think I see what you mean. I thought, at first, that you meant the free ones only resided on the client machine, which means they wouldnt show up on the whole site. But, I see thats not what you meant